Preferential Attachment with Flexible Heavy Tails

Thomas Boughen
t.w.boughen1@newcastle.ac.uk

Newcastle Univeristy

Clement Lee
Clement.Lee@newcastle.ac.uk

Newcastle Univeristy

Vianey Palacios Ramirez
vianey.palacios-ramirez@newcastle.ac.uk

Newcastle Univeristy

June 22, 2025

Mechanistic Models

Preferential attachment

  1. Add vertex
  2. Connect to existing vertex with weights: \[ b(k_i) \] Barabasi-Albert is special case when: \[ b(k) = k+\varepsilon \]

Problems when modelling

  • Scarce evolution data
  • Usually only have a snapshot

Modelling Degrees

Gives information about (in)equality of degrees.

Modelling options

  • Power law
  • Mixture distributions

These don’t give information about how network evolved.

Combining Methods

By using a preference function of the form:

\[ b(k) = \begin{cases} k^\alpha + \varepsilon,&k<k_0\\ k_0^\alpha + \varepsilon + \beta(k-k_0), &k\ge k_0 \end{cases} \]

The survival function for the in-degrees is:

\[ \bar F(k) = \prod_{i=0}^k\frac{b(i)}{\lambda^* + b(i)} \]

Properties

  • Tail heaviness of \(\beta/\lambda^*\)

  • Always heavy tailed

  • Allows super/sub linear behaviour

Testing on Simulated Data

The data

  • 48 networks
  • 48 parameter combinations
  • 100,000 vertex network size

The results

Model are parameters generally recovered quite well.

Results

Modelling Real Data

The data

Modelling Real Data

The results

Fits quite well and performs similarly to a Zipf-IGPD mixture model.

Preference Functions

The bonus

Although the credible interval is quite large, it is still more information than is gained normally.

Conclusions

Outcomes

  • We have presented a viable method for modelling degrees
  • Able to gain information about evolution from degrees alone

Discussion

  • Theory used is for trees, not very applicable to real networks
  • Could include other modifications to model in future e.g. fitness

References